@mmcdara while it is indeed a transfer function, it's the controller's transfer function using the direct synthesis method (not sure about the translation, it's synthèse directe in french). The method works by having the desired form of the reaction of the system to the controller you want to have given, and the transfer function of the process given too, so your only unknown is the controller's transfer function.
Say you want a response of the form 1/(tau_c*s + 1), you would do something like this :
1/(tau_c*s+1) = (Gp*Gc)/(1 + Gp*Gc), where Gp is the process's transfer function and Gc is the controller's.
In the case of this exercice I had been given Gp = ( K*(-b*s + 1)) / (tau^2*s^2 + 2*x*tau*s + 1) and the desired form as (-b*s + 1) / (tau_c*s + 1), with a bit of algebra I obtained the function you can find higher.
Once solving the Gc equation, you get Gc = Kc(1 + 1/(tau_I * s) + tau_D*s), which is the transfer function of a PID controller!
I hope I could explain the context clearly! :)